1d or 2d goniometry method of diffuse sources

ABSTRACT

A goniometry method for one or several diffuse (or distributed) sources is disclosed. The sources or sources having one or more give directions and a diffusion cone. The sources are received by an array of several sensors. The method breaks down the diffusion cone into a finite number L of diffusers. A diffuser has the parameters (θ mp , δθ mpi , Δ mp , δΔ mpi ), associated with it. Directing vectors a(θ mp +δθ mpi , Δ mp +δΔ mpi ) associated with the L diffusers are combined to obtain a vector (D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α)) dependent on at least one of the incidence and deflection parameters (θ, Δ, δθ, δΔ) and on the combination vector α. A MUSIC-type criterion or other goniometry algorithm is applied to the vectors D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α) obtained in order to determine at least one of the incidence parameters θ mp , Δ mp , δθ mp , δΔ mp  of the associated diffusion cone.

The present invention relates to a goniometry method for one or more diffuse, or “distributed”, radiofrequency sources, the source of given direction being considered by the receivers as a diffusion cone with a certain width and an average incidence.

A distributed source is defined notably as a source which is propagated through a continuum of diffusers.

The invention makes it possible notably to locate, in angles and/or in azimuth, one or more distributed radio frequency sources. The object is, for example, to determine the incidence of the centers of the diffusion cones and their widths.

The goniometry is produced either in one dimension, 1D, where the incidences are parameterized by the azimuth, or in two dimensions, 2D, where the incidence depends on both azimuth and elevation parameters.

It applies, for example, for decorrelated or partially decorrelated coherent signals originating from diffusers.

FIG. 1 diagrammatically represents the example of diffusion of the wave from cell phone M through a layer of snow N_(G), for example to the receivers Ci of the reception system of an airplane A. The cone, called diffusion cone, has a certain width and an average incidence. The snow particles N_(G) act as diffusers.

In the field of antenna processing, a multiple-antenna system receives one or more radiocommunication transmitters. The antenna processing therefore uses the signals originating from multiple sensors. In an electromagnetic context, the sensors are antennas. FIG. 2 shows how any antenna processing system consists of an array 1 with several antennas 2 (or individual sensors) receiving the multiple paths from multiple radiofrequency transmitters 3, 4, from different incidence angles and an antenna processing device 5. The term “source” is defined as a multiple path from a transmitter. The antennas of the array receive the sources with a phase and an amplitude dependent on their incidence angle and on the positioning of the antennas. The incidence angles can be parameterized, either in 1D azimuth-wise θ_(m), or in 2D, azimuth-wise θ_(m) and elevation-wise Δ_(m). FIG. 3 shows that a goniometry is obtained in 1D when the waves from the transmitters are propagated in one and the same plane and a 2D goniometry must be applied in other cases. This plane P can be that of the array of antennas where the elevation angle is zero.

The main objective of the antenna processing techniques is to exploit the space diversity, namely, the use of the spatial position of the antennas of the array to make better use of the incidence and distance divergences of the sources. More particularly, the objective of the goniometry or the locating of radiofrequency sources is to estimate the incidence angles of the transmitters from an array of antennas.

Conventionally, the goniometry algorithms such as MUSIC described, for example, in reference [1] (the list of references is appended) assume that each transmitter is propagated according to a discrete number of sources to the listening receivers. The wave is propagated either with a direct path or along a discrete number of multiple paths. In FIG. 2, the first transmitter referenced 3 is propagated along two paths of incidences θ₁₁ and θ₁₂ and the second transmitter referenced 4 along a direct path of incidence θ₂. To estimate the incidences of all of these discrete sources, their number must be strictly less than the number of sensors. For sources that have diffusion cones of non-zero width, the goniometry methods described in document [1] are degraded because of the inadequacy of the signal model.

References [2] [3] [4] propose solutions for the goniometry of distributed sources. However, the proposed goniometry algorithms are in azimuth only: 1D. Also, the time signals of the diffusers originating from one and the same cone are considered to be either coherent in references [2] and [3], or incoherent in references [3] [4]. Physically, the signals of the diffusers are coherent when they are not temporally shifted and have no Doppler shift. Conversely, these signals are incoherent when they are strongly shifted in time or when they have a significant Doppler shift. The time shift of the diffusers depends on the length of the path that the waves follow through the diffusers and the Doppler depends on the speed of movement of the transmitter or of the receivers. These comments show how references [2] [3] [4] do not handle the more common intermediate case of diffusers with partially correlated signals. Also, the algorithms [2] [4] strongly depend on an “a priori” concerning the probability density of the diffusion cones angle-wise. It is then sufficient for these densities to be slightly different from the “a priori” for the algorithms [2] [4] no longer to be suitable.

The subject of the invention concerns notably distributed sources which are received by the listening system in a so-called diffusion cone having a certain width and an average incidence as described for example in FIG. 1.

In this document, the word “source” denotes a multiple-path by diffusion from a transmitter, the source being seen by the receivers in a diffusion cone of a certain width and an average incidence. The average incidence is defined notably by the direction of the source.

The invention relates to a goniometry method for one or several diffuse sources of given directions, the source or sources being characterized by one or more given directions and by a diffusion cone. It is characterized in that it comprises at least the following steps:

-   a) breaking down the diffusion cone into a finite number L of     diffusers, a diffuser having the parameters (θ_(mp), δθ_(mpi),     Δ_(mp), δΔ_(mpi)), associated with it, -   b) combining the directing vectors a(θ_(mp)+δθ_(mpi),     Δ_(mp)+δΔ_(mpi)) associated with the L diffusers to obtain a vector     (D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α)) dependent on at least     one of the incidence and deflection parameters (θ, Δ, δθ, δΔ) and on     the combination vector α, -   c) applying a MUSIC-type criterion or any other goniometry algorithm     to the vectors D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α) obtained in     the step b) in order to determine at least one of the incidence     parameters θ_(mp), Δ_(mp), δθ_(mp), δΔ_(mp) of the associated     diffusion cone.

The minimizing step is, for example, performed on the matrix D(θ, Δ, δθ, δΔ) and implemented according to the parameters θ, Δ, δθ, δΔ.

The minimizing step can be performed on the matrix D_(s)(θ, Δ, δθ, δΔ) where the parameters δθ and/or δΔ are replaced by their opposites.

According to a variant of embodiment, the algorithm comprises a step of limited development of the directing vectors about the central incidence of the cone in order to separate the incidences (θ, Δ) and the deflections δθ, δΔ and in that the minimizing step is performed according to the parameters (θ, Δ) on a matrix U(θ, Δ) dependent on the incidences in order to determine the parameters θ_(mp), Δ_(mp) minimizing the criterion, then secondly to determine the deflection parameters δθ_(mp), δΔ_(mp) from the parameters θ_(mp), Δ_(mp).

The minimizing step is, for example, performed on the matrix U_(s)(θ, Δ) dependent on U(θ, Δ).

The matrix D(θ, δθ) can be dependent only on the azimuth angle θ and on the deflection vector δθ of this angle.

The minimizing step is, for example, performed on the matrix D_(s)(θ, δθ), where the parameter δθ is replaced by its opposite.

The method can include a step of limited development of the vectors of the matrix D(θ, δθ), the minimizing step being performed on a matrix U(θ) in order to determine the incidence angle parameters θ_(mp) and, from these parameters, the angle offset parameters δθ_(mp).

The minimizing step is performed on the matrix U_(s)(θ) dependent on U(θ).

The object of the invention has notably the following advantages:

-   -   producing a goniometry in azimuth and/or in azimuth-elevation,     -   reducing the calculation cost of the method by using a limited         development of the directing vectors,     -   taking into account any type of diffusers, notably         partially-correlated diffusers.

Other characteristics and advantages of the invention will become more apparent from reading the description that follows of an exemplary, nonlimiting embodiment, with appended figures which represent:

FIG. 1, a representation of the diffusion of a wave from a cell phone through a layer of snow,

FIG. 2, an exemplary architecture of an antenna processing system,

FIG. 3, an exemplary 2D goniometry in azimuth-elevation,

FIG. 4, a diagram of the steps of the first variant of the goniometry of distributed sources in azimuth and elevation,

FIG. 5, a diagram of a variant of FIG. 4, taking account of the angular symmetry of the diffusion cones,

FIG. 6, a second variant of the goniometry method for the distributed sources in azimuth and bearing,

FIG. 7, the symmetrical version of the variant of FIG. 6,

FIG. 8, the steps of the first variant of the goniometry in azimuth of the distributed sources,

FIGS. 9 and 10, results of goniometry in azimuth of one or more distributed sources associated with the first variant of the algorithm,

FIG. 11, the symmetrical variant of FIG. 8,

FIG. 12, the steps associated with a second variant of the goniometry in azimuth method,

FIGS. 13 and 14, two results of goniometry in azimuth of distributed sources,

FIG. 15, a diagram of the symmetrical version of the second goniometry in azimuth variant.

In order to better understand the method according to the invention, the description that follows is given, as an illustration and in a nonlimiting way, in the context of the diffusion of the wave from a cell phone through a layer of snow to the receivers on an airplane, for example represented in FIG. 1. The snow particles act as diffusers.

In this example, a diffuse or distributed source is characterized, for example, by a direction and a diffusion cone.

Before detailing the exemplary embodiment, a few reminders are given that may be helpful in understanding the method according to the invention.

General Case

In the presence of M transmitters being propagated along P_(m) non-distributed multiple paths of incidences (θ_(mp), Δ_(mp)) arriving at an array consisting of N sensors, the observation vector x(t) below is received at the output of the sensors:

$\begin{matrix} \begin{matrix} {{x(t)} = \begin{bmatrix} {x_{1}(t)} \\ \vdots \\ {x_{N}(t)} \end{bmatrix}} \\ {= {{\sum\limits_{m = 1}^{M}\; {\sum\limits_{p = 1}^{P_{m}}{\rho_{m\; p}{a\left( {\theta_{m\; p},\Delta_{m\; p}} \right)}{s_{m}\left( {t - \tau_{m\; p}} \right)}^{j\; 2\pi \; {fmp}\; t}}}} + {b(t)}}} \end{matrix} & (1) \end{matrix}$

where x_(n)(t) is the signal received on the nth sensor, a(θ,Δ) is the response from the array of sensors to a source of incidence θ, Δ, s_(m)(t) is the signal transmitted by the mth transmitter, τ_(mp), ƒ_(mp) and ρ_(mp) are respectively the delay, the Doppler shift and the attenuation of the pth multiple path of the mth transmitter and x(t) is the additive noise.

To determine the M_(T)=P₁+ . . . +P_(M) incidences (θ_(mp), Δ_(mp)), the MUSIC method [1] seeks the M_(T) minima ({circumflex over (θ)}_(mp),{circumflex over (Δ)}_(mp)) that cancel the following pseudo-spectrum:

$\begin{matrix} {{{J_{MUSIC}\left( {\theta,\Delta} \right)} = \frac{{a^{H}\left( {\theta,\Delta} \right)}\Pi_{b}{a\left( {\theta,\Delta} \right)}}{{a^{H}\left( {\theta,\Delta} \right)}{a\left( {\theta,\Delta} \right)}}},} & (2) \end{matrix}$

where the matrix Π_(b) depends on the (N−M_(T)) natural vectors e_(MT+i) (1≦i≦N−M_(T)) associated with the lowest natural values of the covariance matrix R_(xx)=E[x(t) x(t)^(H)]: Π_(b)=E_(b)E_(b) ^(H) where E_(b)=[e_(MT+1) . . . e_(N)]. It will also be noted that u^(H) is the conjugate transpose of the vector u. The MUSIC method is based on the fact that the M_(T) natural vectors e_(i) (1≦i≦M_(T)) associated with the highest natural values generate the space defined by the M_(T) directing vectors a(θ_(mp),Δ_(mp)) of the sources such as:

$\begin{matrix} {{e_{i} = {\sum\limits_{m = 1}^{M}\; {\sum\limits_{p = 1}^{P_{m}}{\alpha_{mpi}{a\left( {\theta_{m\; p},\Delta_{m\; p}} \right)}}}}},} & (3) \end{matrix}$

and that the vectors e_(i) are orthogonal to the vectors of the noise space e_(i+MT).

In the presence of M transmitters being propagated along P_(m) distributed multiple paths, the following observation vector x(t) is obtained:

$\begin{matrix} {{x(t)} = {{\sum\limits_{m = 1}^{M}\; {\sum\limits_{p = 1}^{P_{m}}{x_{m\; p}(t)}}} + {b(t)}}} & (4) \end{matrix}$

such that

$\begin{matrix} {{x_{m\; p}(t)} = {\int_{\theta_{m\; p} - {\delta \; \theta_{m\; p}}}^{\theta_{m\; p} + {\delta \; \theta_{m\; p}}}{\int_{\Delta_{m\; p} - {\delta \; \Delta_{m\; p}}}^{\Delta_{m\; p} + {\delta \; \Delta_{m\; p}}}{{\rho \left( {\theta,\Delta} \right)}{a\left( {\theta,\Delta} \right)}{s_{m}\left( {t - {\tau \left( {\theta,\Delta} \right)}} \right)}\ ^{j\; 2\; \pi \; {f{({\theta,\Delta})}}t}{\theta}\ {\Delta}}}}} & \; \end{matrix}$

where (θ_(mp),Δ_(mp)) and (δθ_(mp),δΔ_(mp)) respectively denote the center and the width of the diffusion cone associated with the pth multiple path of the mth transmitter. The parameters τ(θ, Δ), ƒ(θ, Δ) and ρ(θ, Δ) are respectively the delay, the Doppler shift and the attenuation of the diffuser of incidence (θ, Δ). In the presence of coherent diffusers, the delay τ(θ, Δ) and the Doppler shift ƒ(θ, Δ) are zero.

Theory of the Method According to the Invention

The invention is based notably on a breakdown of a diffusion cone into a finite number of diffusers. Using L to denote the number of diffusers of a source, the expression [4] can be rewritten as the following expression [5]:

${x_{m\; p}(t)} = {\sum\limits_{i = 1}^{L}{\rho_{i}{a\left( {{\theta_{m\; p} + {\delta \; \theta_{mpi}}},{\Delta_{m\; p} + {\delta \; \Delta_{mpi}}}} \right)}{s_{m}\left( {t - \tau_{m\; p} - {\delta \; \tau_{mpi}}} \right)}^{j\; 2\; {\pi {({{fmp} + {\delta \; {fmpi}}})}}t}}}$

The expression (5) makes it possible to bring things back to the model of discrete sources (diffusers) of the expression [1] by considering that the individual source is the diffuser of incidence (θ_(mp)+δθ_(mpi), Δ_(mp)+δΔ_(mpi)) associated with the ith diffuser of the pth multiple path of the mth transmitter. In these conditions, the signal space of the covariance matrix R_(xx)=E[x(t) x(t)^(H)] is generated by the vectors a(θ_(mp)+δθ_(mpi), Δ_(mp)+δΔ_(mpi)). By using K to denote the rank of the covariance matrix R_(xx), it can be deduced from this that its natural vectors e_(i) (1≦i≦K) associated with the highest natural values satisfy, according to [3], the following expression:

$\begin{matrix} {e_{i} = {{\sum\limits_{m = 1}^{M}\; {\sum\limits_{p = 1}^{P_{m}}{{c\left( {\theta_{m\; p},\Delta_{m\; p},{\delta \; \theta_{m\; p}},{\delta \; \Delta_{m\; p}},\alpha_{m\; p}^{i}} \right)}\mspace{14mu} {for}\mspace{14mu} 1}}} \leq i \leq K}} & (6) \end{matrix}$

such that

${c\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = {\sum\limits_{j = 1}^{L}{\alpha_{j}{a\left( {{\theta + {\delta \; \theta_{j}}},{\Delta + {\delta \; \Delta_{j}}}} \right)}}}$

with

${{\delta \; \theta} = \begin{bmatrix} {\delta \; \theta_{1}} \\ \vdots \\ {\delta \; \theta_{L}} \end{bmatrix}},{{\delta \; \Delta} = {{\begin{bmatrix} {\delta \; \Delta_{1}} \\ \vdots \\ {\delta \; \Delta_{L}} \end{bmatrix}\mspace{14mu} {and}{\mspace{11mu} \;}\alpha} = \begin{bmatrix} \alpha_{1} \\ \vdots \\ \alpha_{L} \end{bmatrix}}}$

In the presence of coherent diffusers where δτ_(mpi)=0 and δƒ_(mpi)=0, it should be noted that the rank of the covariance matrix satisfies: K=M_(T)=P₁+ . . . +P_(M). In the general case of partially-correlated diffusers where δτ_(mpi)≠0 and δƒ_(mpi)≠0, this rank satisfies K≧M_(T)=P₁+ . . . +P_(M). In the present invention, it is assumed that c(θ_(mp), Δ_(mp), δθ_(mp), δΔ_(mp), α_(mp) ^(i)) is one of the directing vectors associated with the pth multiple path of the mth transmitter and that the unknown parameters are the average incidence (θ_(mp),Δ_(mp)), the angle differences of the diffusers (δθ_(mp), δΔ_(mp)) and one of the vectors α_(mp) ^(i).

Case of Goniometry in Azimuth and in Elevation 1st Variant

FIG. 4 diagrammatically represents the steps implemented according to a first variant of embodiment of the method.

To sum up, the diffusion cone is broken down into L individual diffusers (equation [5]), the different directing vectors a(θ_(mp)+δθ_(mpi), Δ_(mp)+δΔ_(mpi)) are combined, which causes a vector D(θ, Δ, δθ, δΔ) α to be obtained, to which is applied a MUSIC-type or goniometry criterion in order to obtain the four parameters θ_(mp), Δ_(mp), δθ_(mp), δΔ_(mp) which minimize this criterion (the MUSIC criterion is applied to a vector resulting from the linear combination of the different directing vectors).

To determine these parameters with a MUSIC-type algorithm [1], it is essential, according to equations [2] and [6], to find the minima ({circumflex over (θ)}_(mp),{circumflex over (Δ)}_(mp),δ{circumflex over (θ)}_(mp),{circumflex over (α)}_(mp) ^(i)) which cancel the following pseudo-spectrum:

$\begin{matrix} {{{J_{MUSIC\_ diff}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = \frac{\begin{matrix} {c^{H}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} \\ {\Pi_{b}{c\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)}} \end{matrix}}{\begin{matrix} {c^{H}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} \\ {c\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} \end{matrix}}},} & (7) \end{matrix}$

Where the matrix Π_(b) depends on the (N−K) eigenvectors e_(MT+i) (1≦i≦N−K) associated with the lowest natural values of the covariance matrix R_(xx)=E[x(t) x(t)^(H)]: Π_(b)=E_(b) E_(b) ^(H) where E_(b)=[e_(K+1) . . . e_(N)]. Noting, according to the expression [6], that the vector c(θ, Δ, δθ, δΔ, α) can be written in the following form:

c(θ,Δ,δθ,δΔ,α)=D(θ,Δ,δθ,δΔ)α,  (8)

-   -   with D(θ, Δ, δθ, δΔ)=[a(θ+δθ₁, Δ+δΔ₁) . . . a(θ+δθ_(L),         Δ+δΔ_(L))], it is possible to deduce from this that the         criterion J_(MUSIC) _(—) _(diff)(θ, Δ, δθ, δΔ, α) becomes:

$\begin{matrix} {{{J_{MUSIC\_ diff}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = \frac{\alpha^{H}{Q_{1}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta}} \right)}\alpha}{\alpha^{H}{Q_{2}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta}} \right)}\alpha}},} & (9) \end{matrix}$

with Q₁(θ, Δ, δθ, δΔ)=D(θ, Δ, δθ, δΔ)¹¹ Π_(b) D(Oθ, Δ, δθ, δΔ),

and Q₂(θ, Δ, δθ, δΔ)=D(θ, Δ, δθ, δΔ)^(H) D(θ, Δ, δθ, δΔ),

The technique will firstly consist in minimizing the criterion J_(MUSIC) _(—) _(diff)(θ, Δ, δθ, δΔ, α) with α. According to the technique described in reference [2], for example, the criterion J_(min) _(—) _(diff)(θ, Δ, δθ, δΔ) below is obtained:

J _(min) _(—) _(diff)(θ,Δ,δθ,δΔ)=λ_(min) {Q ₁(θ,Δ,δθ,δΔ)Q ₂(θ,Δ,δθ,δΔ)⁻¹}  (10)

where λ_(min)(Q) denotes the minimum natural value of the matrix Q. Noting that the criterion J_(min) _(—) _(diff)(θ, Δ, δθ, δΔ) should be cancelled out for the quadruplets of parameters (θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)) and that det(AB⁻¹)=det(A)/det(B), it can be deduced from this that the quadruplets of parameters (θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)) also cancel the following criterion:

J _(diffision)(θ,Δ,δθ,δΔ)=det(Q ₁(θ,Δ,δθ,δΔ))/det(Q ₂(θ,Δ,δθ,δΔ),  (11)

where det(Q) denotes the determinant of the matrix Q. The M_(T) quadruplets of parameters (θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)) which minimize the criterion J_(diffusion)(θ, Δ, δθ, δΔ) are therefore sought.

FIG. 5 represents the steps of a variant of embodiment taking account of the symmetry of the incidence solution.

Indeed, if (θ,Δ,δθ,δΔ) is the solution, the same applies for (θ,Δ,−δθ,δΔ) (θ,Δ,δθ,−δΔ) (θ,Δ,−δθ,−δΔ). From this comment, it is possible to deduce that:

c(θ,Δ,δθ,δΔ,α)^(H) E _(b)=0,

c(θ,Δ,−δθ,δΔ,α)^(H) E _(b)=0,

c(θ,Δ,δθ,−δΔ,α)^(H) E _(b)=0,

c(θ,Δ,−δθ,−δΔ,α)^(H) E _(b)=0  (11-1)

Where the matrix E_(b) depends on the (N−K) eigenvectors e_(MT+i) (1≦i≦N−K) associated with the lowest eigenvalues of the covariance matrix R_(xx)=E[x(t) x(t)^(H)] such that: E_(b)=[e_(K+1) . . . e_(N)]. From the expression (11-1) it can be deduced that, to estimate the parameters (θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)), it is necessary to find the minima that cancel the following pseudo-spectrum:

$\begin{matrix} {{{{J_{{MUSIC\_ diff}{\_ sym}}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = \frac{\begin{matrix} {c_{s}^{H}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} \\ {\Pi_{bs}{c_{s}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)}} \end{matrix}}{\begin{matrix} {c_{s}^{H}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} \\ {c_{s}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} \end{matrix}}},{{c_{s}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = {\begin{bmatrix} {c\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} \\ {c\left( {\theta,\Delta,{{- \delta}\; \theta},{\delta \; \Delta},\alpha} \right)} \\ {c\left( {\theta,\Delta,{\delta \; \theta},{{- \delta}\; \Delta},\alpha} \right)} \\ {c\left( {\theta,\Delta,{{- \delta}\; \theta},{{- \delta}\; \Delta},\alpha} \right)} \end{bmatrix}\mspace{14mu} {and}}}}{\Pi_{bs} = {\frac{1}{4}E_{bs}E_{bs}^{H}}}{where}{E_{bs} = \begin{bmatrix} E_{b} \\ E_{b} \\ E_{b} \\ E_{b} \end{bmatrix}}} & \left( {11\text{-}2} \right) \end{matrix}$

According to the expression (8), the vector c_(s)(θ, Δ, δθ, δΔ, α) can be written as follows:

$\begin{matrix} {{{{c_{s}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = {{D_{s}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta}} \right)}\alpha}},{with}}{{{D_{s}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta}} \right)} = \begin{bmatrix} {D\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta}} \right)} \\ {D\left( {\theta,\Delta,{{- \delta}\; \theta},{\delta \; \Delta}} \right)} \\ {D\left( {\theta,\Delta,{\delta \; \theta},{{- \delta}\; \Delta}} \right)} \\ {D\left( {\theta,\Delta,{{- \delta}\; \theta},{{- \delta}\; \Delta}} \right)} \end{bmatrix}},}} & \left( {11\text{-}3} \right) \end{matrix}$

The minimizing of J_(MUSIC) _(—) _(diff) _(—) _(sym)(θ, Δ, δθ, δΔ, α) relative to α will lead to the criterion J_(diffusion) _(—) _(sym)(θ, Δ, δθ, δΔ). To obtain J_(diffusion) _(—) _(sym)(θ, Δ, δθ, δΔ), all that is needed is to replace in the expressions (9) (11), D(θ, Δ, δθ, δΔ) with its symmetrical correspondent D_(s)(θ, Δ, δθ, δΔ) and Π_(b) with Π_(bs). The following is thus obtained:

J _(diffusion-sym)(θ,Δ,δθ,δΔ)=det(Q _(1s)(θ,Δ,δθ,δΔ)/det(Q _(2s)(θ,Δ,δθ,δΔ)),  (11-4)

with Q_(1s)(θ, Δ, δθ, δΔ)=D_(s)(θ, Δ, δθ, δΔ)^(H) Π_(bs) D_(s)(θ, Δ, δθ, δΔ),

and Q_(2s)(θ, Δ, δθ, δΔ)=D_(s)(θ, Δ, δθ, δΔ)^(H) D_(s)(θ, Δ, δθ, δΔ),

Therefore, the MT quadruplets of parameters (θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)) which minimize the criterion J_(diffusion-sym)(θ, Δ, δθ, δΔ) are sought.

2nd Variant

FIG. 6 represents a second variant of the goniometry of the diffuse sources in azimuth and elevation that offer notably the advantage of reducing the calculation costs.

The first variant of the goniometry of the sources involves calculating a pseudo-spectrum J_(diffusion) dependent on four parameters (θ, Δ, δθ, δΔ), two of which are vectors of length L. The objective of the second variant is to reduce this number of parameters by performing the limited development along directing vectors about a central incidence (θ, Δ) corresponding to the center of the diffusion cone:

$\begin{matrix} {{a\left( {{\theta + {\delta \; \theta_{i}}},{\Delta + {\delta \; \Delta_{i}}}} \right)} = {{a\left( {\theta,\Delta} \right)} + {\delta \; \theta_{i}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta}} + {\delta \; \Delta_{i}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}} + {\ldots \mspace{14mu} {etc}}}} & (12) \end{matrix}$

where ∂(a(θ, Δ))^(n)/∂θ^(n−p)∂Δ^(p) denotes an nth derivative of the directing vector a(θ,Δ). This corresponds to a limited development about the central incidence (change of base of the linear combination) according to the derivatives of the directing vectors dependent on the central incidence of the cone. From this last expression, it is possible to separate the incidences (θ,Δ) and the deflections (δθ,δΔ) as follows:

$\begin{matrix} {{{a\left( {{\theta + {\delta \; \theta_{i}}},{\Delta + {\delta \; \Delta_{i}}}} \right)} = {{U\left( {\theta,\Delta} \right)}{k\left( {{\delta \; \theta_{i}},{\delta \; \Delta_{i}}} \right)}}}{where}\text{}{{U\left( {\theta,\Delta} \right)} = \left\lbrack {{a\left( {\theta,\Delta} \right)}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}\mspace{14mu} \ldots}\mspace{14mu} \right\rbrack}{and}{{k\left( {{\delta \; \theta_{i}},{\delta \; \Delta_{i}}} \right)} = \begin{bmatrix} 1 \\ {\delta \; \theta_{i}} \\ {\delta \; \Delta_{i}} \\ \vdots \end{bmatrix}}} & (13) \end{matrix}$

According to the expressions (6) (8) and (13), the vector c(θ, Δ, δθ, δΔ, α) becomes:

$\begin{matrix} {{{{c\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = {{U\left( {\theta,\Delta} \right)}{\beta \left( {{\delta \; \theta},{\delta \; \Delta},\alpha} \right)}}},{with}}{{\beta \left( {{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = {\sum\limits_{j = 1}^{L}\; {\alpha_{j}{k\left( {{\delta \; \theta_{i}},{\delta \; \Delta_{j}}} \right)}}}}} & (14) \end{matrix}$

By replacing, in equation (9), D(θ, Δ, δθ, δΔ) with U(θ, Δ) and α with β(δθ, δΔ, α), it is possible to deduce from this, according to (9) (10) (1), that to estimate the M_(T) incidences (θ_(mp), Δ_(mp)) all that is needed is to minimize the following two-dimensional criterion:

J _(diffusion) _(—) _(sym) ^(opt)(θ,Δ)=det(Q ₁ ^(opt)(θ,Δ))/det(Q ₂ ^(opt)(θ,Δ)),  (15)

with

Q₁ ^(opt)(θ, Δ)=U(θ, Δ)^(H) Π_(b) U(θ, Δ) and Q₂ ^(opt)(θ, Δ)=U(θ, Δ)^(H) U(θ, Δ),

Determining the vectors δθ_(mp) and δΔ_(mp) entails estimating the vectors β(δθ_(mp), δΔ_(mp), α). For this, all that is needed is to find the eigenvector associated with the minimum natural value of Q₂ ^(opt) (θ_(mp), Δ_(mp))⁻¹Q₁ ^(opt)(θ_(mp), Δ_(mp)).

FIG. 7 represents a variant of the method of FIG. 6 which takes account of the symmetry of the solutions in order to eliminate some ambiguities. For this, it is important firstly to note that, according to (13) (14):

c(θ,Δ,−δθ,δΔ,α)=U ₁(θ,Δ)β(δθ,δΔ,α)

$\begin{matrix} {{with}\text{}{{{U_{1}\left( {\theta,\Delta} \right)} = \left\lbrack {{a\left( {\theta,\Delta} \right)} - {\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}\mspace{14mu} \ldots}}\mspace{14mu} \right\rbrack},{{c\left( {\theta,\Delta,{\delta \; \theta},{{- \delta}\; \Delta},\alpha} \right)} = {{U_{2}\left( {\theta,\Delta} \right)}{\beta \left( {{\delta \; \theta},{\delta \; \Delta},\alpha} \right)}}}}{with}{{{U_{2}\left( {\theta,\Delta} \right)} = \left\lbrack {{{a\left( {\theta,\Delta} \right)}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta}} - {\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}\mspace{14mu} \ldots}}\mspace{14mu} \right\rbrack},{{c\left( {\theta,\Delta,{{- \delta}\; \theta},{{- \delta}\; \Delta},\alpha} \right)} = {{U_{3}\left( {\theta,\Delta} \right)}{\beta \left( {{\delta \; \theta},{\delta \; \Delta},\alpha} \right)}}}}{with}{{U_{3}\left( {\theta,\Delta} \right)} = \left\lbrack {{a\left( {\theta,\Delta} \right)} - \frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta} - {\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}\mspace{14mu} \ldots}}\mspace{14mu} \right\rbrack}} & \left( {15\text{-}1} \right) \end{matrix}$

From this, according to (11-2), a new expression of the vector c_(s)(θ, Δ, δθ, δΔ, α) can be deduced:

$\begin{matrix} {{{c_{s}\left( {\theta,\Delta,{\delta \; \theta},{\delta \; \Delta},\alpha} \right)} = {{U_{s}\left( {\theta,\Delta} \right)}{\beta \left( {{\delta \; \theta},{\delta \; \Delta},\alpha} \right)}\mspace{14mu} {and}}}{{U_{s}\left( {\theta,\Delta} \right)} = \begin{bmatrix} {U\left( {\theta,\Delta,\alpha} \right)} \\ {U_{1}\left( {\theta,\Delta,\alpha} \right)} \\ {U_{2}\left( {\theta,\Delta,\alpha} \right)} \\ {U_{3}\left( {\theta,\Delta,\alpha} \right)} \end{bmatrix}}} & \left( {15\text{-}2} \right) \end{matrix}$

By replacing, in the equation (15), U_(s)(θ, Δ) with U(θ, Δ), it is possible to deduce from this, according to (11-2) (11-4), that to estimate the M_(T) incidences (θ_(mp),Δ_(mp)) all that is needed is to minimize the following two-dimensional criterion:

J _(diffusion) _(—) _(sym) ^(opt)(θ,Δ)=det(Q _(1s) ^(opt)(θ,Δ))/det(Q _(2s) ^(opt)(θ,Δ)),  (15-3)

with Q_(1s) ^(opt)(θ, Δ)=U_(s)(θ, Δ)^(H) Π_(bs) U_(s)(θ, Δ) and

Q_(2s) ^(opt)(θ, Δ)=U_(s)(θ, Δ)^(H) U_(s)(θ, Δ)

Case of 1D Goniometry in Azimuth

The incidence of a source depends on a single parameter which is the azimuth θ. In these conditions, the directing vector a(θ) is a function of θ. In the presence of M transmitters being propagated along P_(m) distributed multiple paths, the observation vector x(t) of the equation (4) becomes:

${x(t)} = {{\sum\limits_{m = 1}^{M}{\sum\limits_{p = 1}^{P_{m}}{x_{m\; p}(t)}}} + {b(t)}}$

such that

$\begin{matrix} {{x_{m\; p}(t)} = {\int_{\theta_{m\; p} - {\delta \; \theta_{m\; p}}}^{\theta_{m\; p} + {\delta \; \theta_{m\; p}}}{{\rho (\theta)}{a(\theta)}{s_{m}\left( {t - {\tau (\theta)}} \right)}^{j\; 2\; \pi \; {f{(\theta)}}t}\ {\theta}}}} & (16) \end{matrix}$

where θ_(mp) and δθ_(mp) respectively denote the center and the width of the diffusion cone associated with the pth multiple path of the mth transmitter. The parameters τ(θ), ƒ(θ) and ρ(θ) depend only on the azimuth θ of the diffuser. The equation (5) modeling a diffusion cone of a source with L diffusers becomes:

$\begin{matrix} {{x_{m\; p}(t)} = {\sum\limits_{i = 1}^{L}{\rho_{i}{a\left( {\theta_{m\; p} + {\delta \; \theta_{mpi}}} \right)}{s_{m}\left( {t - \tau_{m\; p} - {\delta \; \tau_{mpi}}} \right)}^{j\; 2\; {\pi {({{fmp} + {\delta \; {fmpi}}})}}t}}}} & (17) \end{matrix}$

1st Variant

FIG. 8 diagrammatically represents the steps of the method for the goniometry of the distributed sources in azimuth.

The objective of the 1D goniometry of the distributed sources is to determine the M_(T) doublets of parameters (θ_(mp), δθ_(mp)) which minimize the criterion J_(diffusion)(θ, δθ). It is important to remember that δθ_(mp)=[δθ_(mp1) . . . δθ_(mpL)]^(T), bearing in mind that u^(T) denotes the transpose of u. According to the equations (11), (9) and (8), the criterion J_(diffusion)(θ, δθ) becomes:

J _(diffusion)(θ,δθ)=det(Q ₁(θ,δθ))/det(Q ₂(θ,δθ)),  (18)

with Q₁(θ, δθ)=D(θ, δθ)^(H) Π_(b) D(θ, δθ), ₂(θ, δθ)=D(θ, δθ)^(H) D(θ, δθ), and D(θ, δθ)=[a(θ+δθ₁) . . . a(θ+δθ_(L))]

FIG. 9 simulates the case of a distributed source of average incidence θ₁₁=100° with a cone of width δθ₁₁=20° on a circular array with N=5 sensors of radius R such that R/λ=0.8 (λ denotes the wavelength). In FIG. 9, the method is applied by breaking down the diffusion cone into L=2 diffusers such that δθ=[Δθ−Δθ]^(T). In these conditions, the criterion J_(diffusion)(θ, δθ) depends only on the two scalars θ and Δθ. In this FIG. 9, the function −10 log 10(J_(diffusion)(θ, Δθ)) is plotted, where the maxima correspond to the estimates of the parameters sought.

FIG. 9 shows that the method can well be used to find the center of the diffusion cone in θ₁₁=100° and that the cone going from incidence 80° to 120° is broken down into two paths of incidences θ₁₁−Δθ₁₁=90° and θ₁₁+Δθ₁₁=110°. Bearing in mind that the parameter Δθ₁₁ reflects a barycentric distribution of the diffusers, it is possible to deduce from this that it is necessarily less than the width of the cone δθ₁₁.

FIG. 10, with the same array of sensors, simulates the case of two distributed sources of average incidences θ₁₁=100° and θ₂₂=150° with cones of respective widths δθ₁₁=20° and δθ₂₂=5°. As in the case of the simulation of FIG. 9, the goniometry is applied with L=2 diffusers where δθ=[Δθ−Δθ]^(T).

FIG. 10 shows that the method can be used to estimate with accuracy the centers of the diffusion cones θ₁₁ and θ₂₂ and the parameters Δθ₁₁ and Δθ₂₂ linked to the width of the diffusion cones such that the width of the cone satisfies: δθ_(mp)=2×Δθ_(mp).

FIG. 11 represents the steps of the symmetrical version of the variant described in FIG. 8.

For a goniometry in azimuth, the solution (θ,δθ) necessarily leads to the solution (θ,−δθ). From this comment, it is possible to deduce the following two equations:

c(θ,δθ,α)^(H) E _(b)=0,

c(θ,−δθ,α)^(H) E _(b)=0,

such that, according to (6) (8) (18):

$\begin{matrix} {{{c\left( {\theta,{\delta \; \theta},\alpha} \right)} = {{\sum\limits_{j = 1}^{L}{\alpha_{j}{a\left( {\theta + {\delta \; \theta_{j}}} \right)}}} = {{D\left( {\theta,{\delta \; \theta}} \right)}\alpha}}},} & \left( {18\text{-}1} \right) \end{matrix}$

where the matrix E_(b) depends on the (N−K) eigenvectors e_(MT+i) (1≦i≦N−K) associated with the lowest natural values of the covariance matrix R_(xx)=E[x(t) x(t)^(H)], such that: E_(b)=[e_(K+1) . . . e_(N)]. From the expression (18-1) it can be deduced from this that to estimate the parameters (θ_(mp), δθ_(mp)), it is necessary to search for the minima that cancel the following pseudo-spectrum:

$\begin{matrix} {{{{J_{{MUSIC\_ diff}{\_ sym}}\left( {\theta,{\delta \; \theta},\alpha} \right)} = \frac{{c_{s}^{H}\left( {\theta,{\delta \; \theta},a} \right)}\Pi_{bs}{c_{s}\left( {\theta,{\delta \; \theta},\alpha} \right)}}{{c_{s}^{H}\left( {\theta,{\delta \; \theta},\alpha} \right)}{c_{s}\left( {\theta,{\delta \; \theta},\alpha} \right)}}},{{c_{s}\left( {\theta,{\delta \; \theta},\alpha} \right)} = {\begin{bmatrix} {c\left( {\theta,{\delta \; \theta},\alpha} \right)} \\ {c\left( {\theta,{{- \delta}\; \theta},\alpha} \right)} \end{bmatrix}\mspace{14mu} {and}}}}{\Pi_{bs} = {\frac{1}{2}E_{bs}E_{bs}^{H}\mspace{14mu} {where}}}\text{}{E_{bs} = \begin{bmatrix} E_{b} \\ E_{b} \end{bmatrix}}} & \left( {18\text{-}2} \right) \end{matrix}$

According to the expressions (18-1) (18-2), the vector c_(s)(θ, δθ, α) can be written as follows:

$\begin{matrix} {{{{c_{s}\left( {\theta,{\delta \; \theta},\alpha} \right)} = {{D_{s}\left( {\theta,{\delta \; \theta}} \right)}\alpha}},{with}}{{{D_{s}\left( {\theta,{\delta \; \theta}} \right)} = \begin{bmatrix} {D\left( {\theta,{\delta \; \theta}} \right)} \\ {D\left( {\theta,{{- \delta}\; \theta}} \right)} \end{bmatrix}},}} & \left( {18\text{-}3} \right) \end{matrix}$

The minimizing of J_(MUSIC) _(—) _(diff) _(—) _(sym)(θ, δθ, α) relative to α will lead to the criterion J_(diffusion) _(—) _(sym)(θ, δθ). To obtain J_(diffusion) _(—) _(sym)(θ,δθ), all that is needed is to replace in expression (18), D(θ,δθ) with D_(s)(θ,δθ) and Π_(b) with Π_(bs). The following is thus obtained:

J _(diffusion-sym)(θ,δθ)=det(Q _(1s)(θ,δθ))/det(Q _(2s)(θ,δθ)),  (18-4)

with Q_(1s)(θ, δθ)=D_(s)(θ, δθ)^(H) Π_(bs) D_(s)(θ, δθ) and Q_(2s)(θ, δθ)=D_(s)(θ,δθ)^(H) D_(s)(θ, δθ)

The M_(T) doublets of parameters (θ_(mp), δθ_(mp)) that minimize the criterion J_(diffusion-sym)(θ,δθ) are therefore sought.

2nd Variant

FIG. 12 diagrammatically represents the steps of the second variant of the goniometry of the distributed sources in azimuth.

By performing a limited development of the order I of a(θ+δθ_(i)) about the central incidence θ, the expression (13) becomes as follows:

$\begin{matrix} {{{a\left( {\theta + {\delta \; \theta_{i}}} \right)} = {{U(\theta)}{k\left( {\delta \; \theta_{i}} \right)}}}{where}{{U(\theta)} = \left\lbrack {{a(\theta)}\frac{\partial{a(\theta)}}{\partial\theta}\mspace{14mu} \ldots \mspace{14mu} \frac{\partial\left( {a(\theta)} \right)^{I}}{\partial\theta^{I}}} \right\rbrack}{and}{{k\left( {\delta \; \theta_{i}} \right)} = \begin{bmatrix} 1 \\ {\delta \; \theta_{i}} \\ \vdots \\ \frac{\delta \; \theta_{i}^{I}}{I!} \end{bmatrix}}} & (19) \end{matrix}$

It can be deduced from this that the vector c(θ, δθ, α) of the expression (18-1) is written, according to (14):

$\begin{matrix} {{{c\left( {\theta,{\delta \; \theta},\alpha} \right)} = {{U(\theta)}{\beta \left( {{\delta \; \theta},\alpha} \right)}}}{with}{{{\beta \left( {{\delta \; \theta},\alpha} \right)} = {\sum\limits_{j = 1}^{L}{\alpha_{j}{k\left( {\delta \; \theta_{j}} \right)}}}},}} & \left( {19\text{-}1} \right) \end{matrix}$

The aim of the second variant of the 1D goniometry of diffuse sources is to determine the M_(T) incidences θ_(mp) that minimize the criterion J_(difftsion) ^(opt)(θ). According to the equations (15) and (14), the criterion J_(diffusion) ^(opt)(θ) becomes:

J _(diffusion) ^(opt)(θ)=det(Q ₁ ^(opt)(θ))/det(Q ₂ ^(opt)(θ)),  (20)

with Q₁ ^(opt)(θ)=U(θ)^(H) Π_(b) U(θ) and Q₂ ^(opt)(θ)=U(θ)^(H) U(θ), and

${{\beta \left( {{\delta \; \theta},\alpha} \right)} = {\sum\limits_{j = 1}^{L}{\alpha_{j}{k\left( {\delta \; \theta_{j}} \right)}}}},$

Determining the vectors δθ_(mp) entails estimating the vector δ(δθ_(mp), α): For this, all that is needed is to find the eigenvector associated with the minimum natural value of Q₂ ^(opt)(θ_(mp))⁻¹ Q₁ ^(opt)(θ_(mp)).

In FIG. 13, the MUSIC performance levels are compared with those of the second variant of distributed MUSIC for I=1 and I=2. The array of sensors is that of FIG. 9 and of FIG. 10. The case of two distributed sources of average incidence θ₁₁=100° and θ₂₂=120° with cones of respective width δθ₁₁=20° and δθ₂₂=20° is simulated. It should be remembered that the M_(T) maxima of the function −10 log 10(J_(diffusion) ^(opt)(θ)) are the estimates of the incidences θ_(mp) sought. The curves of FIG. 13 show that the more the order I of the limited development increases, the higher the level of the two maxima of the criterion becomes, because there is a convergence towards a good approximation of the model. Table 1 gives the estimates of the incidences for the three methods.

TABLE 1 Goniometry in azimuth of a distributed source (θ₁₁ = 100° θ₂₂ = 120° with a cone of width δθ₁₁ = 20° δθ₂₂ = 20°) with the second variant {circumflex over (θ)}₁₁ given that {circumflex over (θ)}₂₂ given that θ₁₁ = 100° θ₂₂ = 120° Conventional 97 123.1 MUSIC (I = 0) Distributed 101.7 118 MUSIC (I = 1) Distributed 99.2 120.6 MUSIC (I = 2)

Table 1 confirms that the lowest incidence estimation bias is obtained for I=2, that is, for the distributed MUSIC method of order of interpolation of the highest directing vector interpolation order. The simulation of FIG. 10 and of table 1 is obtained for a time spread of the two zero sources. More precisely, the delays δτ_(11i) and δτ_(22i) of (17) of the diffusers are zero. In the simulation of table 2 and of FIG. 11, the preceding configuration is retained, but with a time spread of one sampling period T_(e) introduced such that: max (δτ_(mmi))−min (δτ_(mmi))=T_(e)

TABLE 2 Goniometry in azimuth of partially correlated distributed sources θ₁₁ = 100° θ₂₂ = 120° with a cone of width δθ₁₁ = 20° δθ₂₂ = 20° with the 2nd variant {circumflex over (θ)}₁₁ given that {circumflex over (θ)}₂₂ given that θ₁₁ = 100° θ₂₂ = 120° Conventional 96.2 124.3 MUSIC (I = 0) Distributed 98.1 121.5 MUSIC (I = 2)

The results of Table 2 and of FIG. 14 show that the methods envisaged in this invention take into account the configurations of partially-correlated diffusers

In this second variant it is possible, as in the first variant, to take into account the symmetry of the solutions in order to eliminate some ambiguities. For this, it should first be noted that, according to (19) (19-1):

$\begin{matrix} {{{c\left( {\theta,{{- \delta}\; \theta},\alpha} \right)} = {{U_{1}(\theta)}{\beta \left( {{\delta \; \theta},\alpha} \right)}}}{with}{{U_{1}(\theta)} = \left\lbrack {{a(\theta)} - {\frac{\partial{a(\theta)}}{\partial\theta}\mspace{14mu} \ldots \mspace{14mu} \left( {- 1} \right)^{I}\frac{\partial\left( {a(\theta)} \right)^{I}}{\partial\theta^{I}}}} \right\rbrack}} & \left( {20\text{-}1} \right) \end{matrix}$

According to (18-1) (18-2), a new expression of the vector c_(s)(θ, δθ, α) can be deduced:

$\begin{matrix} {{{c_{s}\left( {\theta,{\delta \; \theta},\alpha} \right)} = {{U_{s}(\theta)}{\beta \left( {{\delta \; \theta},\alpha} \right)}}}{and}{{U_{s}(\theta)} = \begin{bmatrix} {U\left( {\theta,\alpha} \right)} \\ {U_{1}\left( {\theta,\alpha} \right)} \end{bmatrix}}} & \left( {20\text{-}2} \right) \end{matrix}$

By replacing, in the equation (20) U_(s)(θ) with U(θ) and Π_(b) with Π_(bs), it can be deduced from this, according to (18-2) (18-4), that to estimate the M_(T) incidences (θ_(mp)), all that is needed is to minimize the following one-dimensional criterion:

J _(diffusion) _(—) _(sym) ^(opt)(θ)=det(Q _(1s) ^(opt)(θ))/det(Q _(2s) ^(opt)(θ)),  (20-3)

with Q_(1s) ^(opt)(θ)=U_(s)(θ)^(H) Π_(bs) U_(s)(θ) and Q_(2s) ^(opt)(θ)=U_(s)(θ)^(H) U_(s)(θ),

This symmetrical version of the second variant of the goniometry of distributed sources in azimuth is summarized in FIG. 15.

-   [1] RO. SCHMIDT “A signal subspace approach to multiple emitter     location and spectral estimation”, PhD thesis, Stanford University     CA, November 1981. -   [2] FERRARA, PARKS “Direction finding with an array of antennas     having diverse polarizations”, IEEE trans on antennas and     propagation, March 1983. -   [3] S. VALAE, B. CHAMPAGNE and P. KABAL “Parametric Localization of     Distributed Sources”, IEEE trans on signal processing, Vol 43 no 9     September 1995. -   [4] D. ASZTELY, B. OTTERSTEN and AL. SWINDLEHURST “A Generalized     array manifold model for local scattering in wireless     communications”, Proc of ICASSP, pp 4021-4024, Munich 1997. -   [5] M. BENGTSSON and B. OTTERSTEN “Low-Complexity Estimators for     Distributed Sources”, trans on signal processing, vol 48, no 8,     August 2000. 

1. A goniometry method for one or several distributed sources, the source or sources having one or more given directions and by a diffusion cone and received by an array of several sensors, comprising the following steps: a) breaking down the diffusion cone into a finite number L of diffusers, a diffuser having the parameters (θ_(mp), δθ_(mpi), Δ_(mp), δΔ_(mpi)), associated with it, b) combining the directing vectors a(θ_(mp)+δθ_(mpi), Δ_(mp)+δΔ_(mpi)) associated with the L diffusers to obtain a vector (D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α)) dependent on at least one of the incidence and deflection parameters (θ, Δ, δθ, δΔ) and on the combination vector α, c) applying a MUSIC-type criterion or any other goniometry algorithm to the vectors D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α) obtained in the step b) in order to determine at least one of the incidence parameters θ_(mp), Δ_(mp), δθ_(mp), δΔ_(mp) of the associated diffusion cone.
 2. The goniometry method as claimed in claim 1, wherein the minimizing step is performed on the matrix D(θ, Δ, δθ, δΔ) and implemented according to the parameters θ, Δ, δθ, δΔ.
 3. The goniometry method as claimed in claim 2, wherein the minimizing step is performed on the matrix D_(s)(θ, Δ, δθ, δΔ) where the parameters δθ and/or δΔ are replaced by their opposites.
 4. The goniometry method as claimed in claim 1, wherein it comprises a step of limited development of the directing vectors about the central incidence of the cone in order to separate the incidences (θ, Δ) and the deflections δθ, δΔ and in that the minimizing step is performed according to the parameters (θ, Δ) on a matrix U(θ, Δ) dependent on the incidences in order to determine the parameters θ_(mp), Δ_(mp) minimizing the criterion, then secondly to determine the deflection parameters δθ_(mp), δΔ_(mp) from the parameters θ_(mp), Δ_(mp).
 5. The goniometry method as claimed in claim 4, wherein the minimizing step is performed on the matrix U_(s)(θ, Δ) dependent on U(θ, Δ).
 6. The goniometry method as claimed in claim 1, wherein the matrix D(θ, δθ) depends only on the azimuth angle θ and on the deflection vector δθ of this angle.
 7. The goniometry method as claimed in claim 6, wherein the minimizing step is performed on the matrix D_(s)(θ, δθ), where the parameter δθ is replaced by its opposite.
 8. The goniometry method as claimed in claim 6, wherein it comprises a step of limited development of the vectors of the matrix D(θ, δθ), the minimizing step being performed on a matrix U(θ) in order to determine the incidence angle parameters θ_(mp) and, from these parameters, the angle offset parameters δθ_(mp).
 9. The goniometry method as claimed in claim 7, wherein the minimizing step is performed on the matrix U_(s)(θ) dependent on U(θ).
 10. The goniometry method as claimed in claim 8, wherein the minimizing step is performed on the matrix U_(s)(θ) dependent on U(θ). 